Internal problem ID [9141]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1566.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _with_linear_symmetries]]
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+6 y^{\prime \prime \prime } x^{3}+\left (4 x^{4}+\left (-2 \mu ^{2}-2 \nu ^{2}+7\right ) x^{2}\right ) y^{\prime \prime }+\left (16 x^{3}+\left (-2 \mu ^{2}-2 \nu ^{2}+1\right ) x \right ) y^{\prime }+\left (8 x^{2}+\left (\mu ^{2}-\nu ^{2}\right )^{2}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 37
dsolve(x^4*diff(diff(diff(diff(y(x),x),x),x),x)+6*x^3*diff(diff(diff(y(x),x),x),x)+(4*x^4+(-2*mu^2-2*nu^2+7)*x^2)*diff(diff(y(x),x),x)+(16*x^3+(-2*mu^2-2*nu^2+1)*x)*diff(y(x),x)+(8*x^2+(mu^2-nu^2)^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {BesselJ}\left (\nu , x\right ) \operatorname {BesselJ}\left (\mu , x\right )+c_{2} \operatorname {BesselJ}\left (\nu , x\right ) \operatorname {BesselY}\left (\mu , x\right )+c_{3} \operatorname {BesselY}\left (\nu , x\right ) \operatorname {BesselJ}\left (\mu , x\right )+c_{4} \operatorname {BesselY}\left (\nu , x\right ) \operatorname {BesselY}\left (\mu , x\right ) \]
✓ Solution by Mathematica
Time used: 0.326 (sec). Leaf size: 95
DSolve[((\[Mu]^2 - \[Nu]^2)^2 + 8*x^2)*y[x] + ((1 - 2*\[Mu]^2 - 2*\[Nu]^2)*x + 16*x^3)*y'[x] + ((7 - 2*\[Mu]^2 - 2*\[Nu]^2)*x^2 + 4*x^4)*y''[x] + 6*x^3*Derivative[3][y][x] + x^4*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2^{-\mu -\nu } \left (\operatorname {Gamma}(1-\mu ) \operatorname {BesselJ}(-\mu ,x) \left (c_1 \operatorname {Gamma}(1-\nu ) \operatorname {BesselJ}(-\nu ,x)+c_3 4^{\nu } \operatorname {Gamma}(\nu +1) \operatorname {BesselJ}(\nu ,x)\right )+4^{\mu } \operatorname {Gamma}(\mu +1) \operatorname {BesselJ}(\mu ,x) \left (c_2 \operatorname {Gamma}(1-\nu ) \operatorname {BesselJ}(-\nu ,x)+c_4 4^{\nu } \operatorname {Gamma}(\nu +1) \operatorname {BesselJ}(\nu ,x)\right )\right ) \\ \end{align*}